JournalsrmiVol. 32, No. 3pp. 751–794

Global wellposedness of the equivariant Chern–Simons–Schrödinger equation

  • Baoping Liu

    Peking University, Beijing, China
  • Paul Smith

    University of California Berkeley, USA
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Abstract

In this article we consider the initial value problem for the mm-equivariant Chern–Simons–Schrödinger model in two spatial dimensions with coupling parameter gRg \in \mathbb R. This is a covariant NLS type problem that is L2L^2-critical. We prove that at the critical regularity, for any equivariance index mZm \in \mathbb Z, the initial value problem in the defocusing case (g<1g < 1) is globally wellposed and the solution scatters. The problem is focusing when g1g \geq 1, and in this case we prove that for equivariance indices mZm \in \mathbb Z, m0m \geq 0, there exist constants c=cm,gc = c_{m, g} such that, at the critical regularity, the initial value problem is globally wellposed and the solution scatters when the initial data ϕ0L2\phi_0 \in L^2 is mm-equivariant and satisfies ϕ0L22<cm,g\| \phi_0 \|_{L^2}^2 < c_{m, g}. We also show that cm,g\sqrt{c_{m, g}} is equal to the minimum L2L^2 norm of a nontrivial mm-equivariant standing wave solution. In the self-dual g=1g = 1 case, we have the exact numerical values cm,1=8π(m+1)c_{m, 1} = 8\pi(m + 1).

Cite this article

Baoping Liu, Paul Smith, Global wellposedness of the equivariant Chern–Simons–Schrödinger equation. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 751–794

DOI 10.4171/RMI/898